Symmetry operators in for second quantized operators only act on second quantized operators and do not act on numbers (i.e. the ‘first quantized Hamiltonian’, , or Hamiltonian density), UNLESS the symmetry operator is the time reversal operator, .

The symmetry operation, , is defined in terms of second quantized field operators: , where is a unitary matrix.

If a second quantized Hamiltonian, , is invariant under the symmetry operator, , then we have . By using the way the symmetry operation acts on the field operators, we can define an effective action of the symmetry operator on the Hamiltonian density, .

According to the above comment, it is trivial to write something like since we always have , unless when is the time reversal operator , under which circumstance .

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